Consider the article:
http://mathworld.wolfram.com/FieldCharacteristic.html
It is stated that given a field and its multiplicative identity $I_{\times}$ that either:
$$ \sum_{i=0}^{k}{I_{\times}} $$
Is unique for all k. OR... there exists A and B such that
$$ \sum_{i=0}^{A}{I_{\times}} = \sum_{i=0}^{P}{I_{\times}} , P > A \ge 0 $$
So I'm totally down with that business...
Here's what I'm not down with,
"The smallest such P must be a prime".
Why on earth is that true?
Consider the field of Integers modulo $10$. Smallest P is 10 which isn't a prime at all.
Note:
$$ 1 + 1 + 1 ... 1 \equiv 0 \mod 10 $$
$$ \{ empty \ sum \} \equiv 0 \mod 10 $$
Clearly i'm missing something. What is it?
The fact that the characteristic of a field is either $0$ or prime (which is essentially what you are saying) is well known and can be found in any text. In fact, just try to prove it as an exercise. To clarify your confusion, the integers modulo $10$ do not form a field (not every element has a multiplicative inverse).